Large time behavior in a predator-prey system with indirect pursuit-evasion interaction

By Genglin Li, Youshan Tao, Michael Winkler. 2017

48 citations recorded by the source corpus.

In a bounded domain \begin{document}$ \Omega\subset \mathbb{R}^n $\end{document} with smooth boundary, this work considers the indirect pursuit-evasion model \begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{l} u_t = \Delta u -\chi\nabla \cdot(u \nabla w) +u(\lambda -u+a v), \\ v_t = \Delta v +\xi \nabla \cdot(v\nabla z) +v(\mu-v-b u), \\ 0 = \Delta w -w +v, \\ 0 = \Delta z -z +u, \end{array} \right. \end{eqnarray*} $\end{document} with positive parameters \begin{document}$ \chi, \xi, \lambda, \mu $\end{document} , \begin{document}$ a $\end{document} and \begin{document}$ b $\end{document} . It is firstly asserted that when \begin{document}$ n\le 3 $\end{document} , for any given suitably regular initial data the corresponding homogeneous Neumann initial-boundary problem admits a global and bounded smooth solution. Moreover, it is shown that if \begin{document}$ b\lambda and under some explicit smallness conditions on \begin{document}$ \chi $\end{document} and \begin{document}$ \xi $\end{document} , any nontrival bounded classical solution converges to the spatially homogeneous coexistence state in the large time limit; if \begin{document}$ b\lambda>\mu $\end{document} , however, then under an explicit smallness assumption on \begin{document}$ \chi $\end{document} but without any restriction on \begin{document}$ \xi $\end{document} , any bounded classical solution \begin{document}$ (u, v) $\end{document} with \begin{document}$ u\not\equiv 0 $\end{document} stabilizes to \begin{document}$ (\lambda, 0) $\end{document} as \begin{document}$ t\to \infty $\end{document} .

Subfields: Genetics, Modeling and Simulation, Public Health, Environmental and Occupational Health

View Paper: https://www.semanticscholar.org/paper/b07e2dfcbc160b33d97c97c6f941f78d92add499